30 research outputs found
Fixed points of the EM algorithm and nonnegative rank boundaries
Mixtures of independent distributions for two discrete random variables
can be represented by matrices of nonnegative rank . Likelihood inference
for the model of such joint distributions leads to problems in real algebraic
geometry that are addressed here for the first time. We characterize the set of
fixed points of the Expectation-Maximization algorithm, and we study the
boundary of the space of matrices with nonnegative rank at most . Both of
these sets correspond to algebraic varieties with many irreducible components.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1282 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Maximum likelihood estimation of toric Fano varieties
We study the maximum likelihood estimation problem for several classes of
toric Fano models. We start by exploring the maximum likelihood degree for all
-dimensional Gorenstein toric Fano varieties. We show that the ML degree is
equal to the degree of the surface in every case except for the quintic del
Pezzo surface with two ordinary double points and provide explicit expressions
that allow one to compute the maximum likelihood estimate in closed form
whenever the ML degree is less than 5. We then explore the reasons for the ML
degree drop using -discriminants and intersection theory. Finally, we show
that toric Fano varieties associated to 3-valent phylogenetic trees have ML
degree one and provide a formula for the maximum likelihood estimate. We prove
it as a corollary to a more general result about the multiplicativity of ML
degrees of codimension zero toric fiber products, and it also follows from a
connection to a recent result about staged trees.Comment: 28 pages, 4 figures, 4 tables, this article supersedes
arXiv:1602.0830
Model embeddability for symmetric group-based models
We study model embeddability, which is a variation of the famous embedding problem in probability theory, when apart from the requirement that the Markov matrix is the matrix exponential of a rate matrix, we additionally ask that the rate matrix follows the model structure. We provide a characterisation of model embeddable Markov matrices corresponding to symmetric group-based phylogenetic models. In particular, we provide necessary and sufficient conditions in terms of the eigenvalues of symmetric group-based matrices. To showcase our main result on model embeddability, we provide an application to hachimoji models, which are eight-state models for synthetic DNA. Moreover, our main result on model embeddability, enables us to compute the volume of the set of model embeddable Markov matrices relative to the volume of other relevant sets of Markov matrices within the model
Algebraic boundary of matrices of nonnegative rank at most three
The Zariski closure of the boundary of the set of matrices of nonnegative
rank at most 3 is reducible. We give a minimal generating set for the ideal of
each irreducible component. In fact, this generating set is a Grobner basis
with respect to the graded reverse lexicographic order. This solves a
conjecture by Robeva, Sturmfels and the last author.Comment: 15 pages, 2 figure
On the uniqueness of collections of pennies and marbles
In this note we study the uniqueness problem for collections of pennies and
marbles. More generally, consider a collection of unit -spheres that may
touch but not overlap. Given the existence of such a collection, one may
analyse the contact graph of the collection. In particular we consider the
uniqueness of the collection arising from the contact graph. Using the language
of graph rigidity theory, we prove a precise characterisation of uniqueness
(global rigidity) in dimensions 2 and 3 when the contact graph is additionally
chordal. We then illustrate a wide range of examples in these cases. That is,
we illustrate collections of marbles and pennies that can be perturbed
continuously (flexible), are locally unique (rigid) and are unique (globally
rigid). We also contrast these examples with the usual generic setting of graph
rigidity.Comment: 9 pages, 11 figure